Base field 6.6.1387029.1
Generator \(a\), with minimal polynomial \( x^{6} - 3 x^{5} - 2 x^{4} + 9 x^{3} - x^{2} - 4 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -4, -1, 9, -2, -3, 1]))
gp: K = nfinit(Polrev([1, -4, -1, 9, -2, -3, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, -1, 9, -2, -3, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1,-3,-1,1,0]),K([0,3,0,-1,0,0]),K([2,0,-7,2,3,-1]),K([132,-349,-610,353,218,-96]),K([-757,1985,3487,-2010,-1250,551])])
gp: E = ellinit([Polrev([1,1,-3,-1,1,0]),Polrev([0,3,0,-1,0,0]),Polrev([2,0,-7,2,3,-1]),Polrev([132,-349,-610,353,218,-96]),Polrev([-757,1985,3487,-2010,-1250,551])], K);
magma: E := EllipticCurve([K![1,1,-3,-1,1,0],K![0,3,0,-1,0,0],K![2,0,-7,2,3,-1],K![132,-349,-610,353,218,-96],K![-757,1985,3487,-2010,-1250,551]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^5-3a^4-2a^3+8a^2+a-2)\) | = | \((a^5-3a^4-2a^3+8a^2+a-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 3 \) | = | \(3\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-a^4+2a^3+5a^2-6a-2)\) | = | \((a^5-3a^4-2a^3+8a^2+a-2)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 6561 \) | = | \(3^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{1737352}{81} a^{4} - \frac{3474704}{81} a^{3} - \frac{8452517}{81} a^{2} + \frac{3396623}{27} a + \frac{10114849}{81} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Mordell-Weil group
| Rank: | \(0\) |
| Torsion structure: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(3 a^{5} - 7 a^{4} - 11 a^{3} + 20 a^{2} + 10 a - 4 : a^{5} - 2 a^{4} - 2 a^{3} + 4 a^{2} - 1 : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 2794.5449739841247883070991341980285239 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.18642 \) | ||
| Analytic order of Ш: | \( 1 \) (rounded) | ||
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
| prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^5-3a^4-2a^3+8a^2+a-2)\) | \(3\) | \(2\) | \(I_{8}\) | Non-split multiplicative | \(1\) | \(1\) | \(8\) | \(8\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
3.1-a
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| 3.3.257.1 | a curve with conductor norm 147 (not in the database) |
| 3.3.257.1 | 3.3.257.1-9.2-a4 |